On a Degenerate Heat Equation with a Singular Potential

Abstract

Using a new method, we generalize the blow up and existence result from P. Baras and J. A. Goldstein (1984, Trans. Amer. Math. Soc.284, 121–139) to heat equations on the Heisenberg group. In doing so we need to overcome the difficulty that the equation in this case is both degenerate and of variable coefficients. Comparing with the Euclidean case, an interesting new result is that solutions can blow up even when the singularity of the potential is weaker than the inverse square of the distance function.